Nielsen theory of roots of maps of pairs

TOPOLOGY ANDITS APPLICATIONS EL..WVIER

Topology

and its Applications 92 (1999) 247-274

Nielsen theory of roots of maps of pairs Robert F. Brown al*, Helga Schirmer b a University of California,

Los Angeles, CA 90095-1555, USA b Carleton University, Ottawa, Ontario KIS .5B6, Canada

Received 8 August 1997; received in revised form 26 September 1997

Abstract A relative root Nielsen number IV,&; c ) is introduced which is a homotopy invariant lower bound for the number of roots at c for a map of pairs of spaces f : (X, A) + (Y, B) and c E Y. Conditions are given which ensure that N,l( f; c) is a sharp lower bound. The standard method for the computation of the root Nielsen number N(f; c) from me homomorphism of the fundamental group induced by f is extended to obtain formulae for N&f; c). Many of me results depend on the location of c. 0 1999 Elsevier Science B.V. All rights reserved. Keywords: Nielsen numbers; Relative root Nielsen number; Relatively essential; Multiplicity; Boundary-preserving map; Minimum number of roots; Computation of root Nielsen numbers; Degree of a map AMS classijication:

Primary 55M20, Secondary 57N99; 54825

1. Introduction Let f : X + that is a solution

Y be a map and c E Y a point. A root of f at c is a point x E X to the equation

f(x)

= c. Denote the set of roots by root(f; c) and

let #root(f; c) be the cardinality of that set. (Throughout this paper, we will indicate the cardinality of a set 5’ by #S.) Nielsen root theory is concerned with A4%[f; c] = min {#root(g; c): g N f}, where g - f means that the minimum is taken over all maps g homotopic to f. Influenced by Nielsen fixed point theory, Hopf [ 121 introduced such a theory in 1930, as he found it to be a natural and very helpful tool in order to clarify the relation between the algebraic and geometric definitions of the degree of a map between two manifolds * Corresponding author. E-mail: [email protected]. 0166-8641/99/$ - see front matter 0 1999 Elsevier Science B.V. All tights reserved. PII: SOl66-8641(97)00248-4

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R.E Brown, H. Schirmer / Topology and its Applications 92 (1999) 247-274

of the same dimension. introduction

But while Hopf’s contribution

of Nielsen root theory was forgotten

ject was rediscovered

independently,

to degree theory survived,

his

for almost forty years, until the sub-

and in a more general setting, by Brooks, starting

with [l]. Brooks did not study the relation of Nielsen root theory to degree theory, but instead introduced is concerned

root theory as part of Nielsen coincidence

with solutions

so the root equation

f(z)

to the equation

f(z)

= c can be identified

theory. Coincidence

theory

= g(z) for pairs of maps f, g : X -+ Y, with the case in which g is the constant

map at c E Y. Some aspects of Nielsen root theory are simpler than the corresponding ones in Nielsen coincidence theory and there are many special properties of root theory that cannot be obtained from coincidence theory. By analogy with Nielsen fixed point and coincidence theory, a Nielsen number of roots N(f;

c) was defined by Hopf and by Brooks which is a lower bound for MR[f; c]. Hopf

showed that there exist maps f between closed oriented surfaces for which N(f; c) is strictly less than MR[f; c] but if f : X + Y is a map between closed oriented nmanifolds with n # 2, then N(f;c) = MR[f; c1, w h’ic h means that N(f; c) is a sharp lower bound for the least number of roots at c for all maps in the homotopy class of f. The computation of N(f; c) depends only on information about the map f itself, and not on the entire homotopy

class as MR[f; c] does. The number

N(f;

c) can often

be computed easily just from a knowledge of the homomorphism induced by f on the fundamental groups, and the conditions required for this computation are surprisingly weak, at least when compared to the corresponding result for the Nielsen number in fixed point theory. Thus the two basic problems of any Nielsen type theory, establishing the sharpness

of the Nielsen

number

as a lower bound and obtaining

methods

for its

computation, have been solved to a large extent in Nielsen root theory. References to these solutions are given later in this paper. Nielsen fixed point theory has been extended to the setting of maps of pairs in [21] and many subsequent papers (see [22] for a survey). Given a map of pairs f : (X, A) ---) (X, A), this theory is concerned with the minimum number of fixed points among all maps g: (X,A) + (X,A) that are homotopic, as maps of pairs, to f. The purpose of this paper is to investigate the root concept in the setting of maps of pairs. Thus, we begin with a map of pairs f : (X,A) -+ (Y,B) and we recall that g:(X,A) -+ (Y,B) is h omotopic to f as a map of pairs if there is a map of pairs H:(X x I,A x I) -+ (Y, B) such that H(z, 0) = f(z) and H(z, 1) = g(z) for all 2 E X. We will indicate that maps f and g are homotopic as maps of pairs by writing f M g, We still have a point c E Y. In the setting of maps of pairs, instead minimum

number

MR,l[f;

MR[f; c] we are concerned

of the

with

c] = min {#root(g, c): g M f}.

By analogy with the theory of Hopf and Brooks, we will define a lower bound N,l(f; for MR,Jf; c], find conditions for the equality %a(f;

c) = M&[f;

to hold, and investigate

cl conditions

under which N,l(f;

c) is readily computable.

c)

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R.E Brown, H. Schimer / Topology and its Applications 92 (1999) 247-274

Although

it is easy to see in broad outline what features a relative root theory should

have, the following

example demonstrates

of maps of pairs is not entirely

that the extension

of root theory to the setting

straightforward.

Example 1.1. Let X = Y = D be the unit disc in the complex plane and let A = B = aD be the boundary

circle. Writing a point z E D in the form 2 = reie for 0 < T < 1

and 0 < 8 < 2x, choose a nonzero integer d and define f : (D, 3D) + (D, 3D) by f(re”) = redi’. Denote by f: aD + aD the restriction of f. We choose c = 0. We have MR,l[f;

c] = 1 because the only solution to f(x)

:= 0 is z = 0 and MR,l[f;

On the other hand, of course MR[f; c] = 0 because the map f : D 4

c] # 0.

D is homotopic

to a constant map, so N(f; c) = 0 as well and we are not surprised to discover that the Nielsen number of roots N(f; c ) IS not adequate for studying roots of maps of pairs. But now suppose we choose c to be a point on a D, say c = #root(fic) = jdl. By [16, Chapter V, Example 3.3, p. N(f;c) = IdJ < MR[f; c] an d since N(f; c) < #root(g; that MR,l[f; c] = IdI in th’is case. This example shows

1. We observe that #root( f; c) = 127 and Corollary 7.3, p. 1381, c) for any g M f, we conclude us that the value of MR,l[p; c]

can depend on whether or not c E B, so our Nielsen theory must take this feature of the behavior of roots of maps of pairs into account. In Section 2, we will define the Nielsen number N,,l(f; The definition

has the same structure

as the relative

c) of roots of a map of pairs.

Nielsen

number

of [21], but we

will see that, in the theory of roots, we need to adapt the key concept of essential root class (see Section 2) to the setting of maps of pairs by introducing a notion of relative essentiality. We will prove in that section that Nrel(f; c) is homotopy invariant and that it is a lower bound for MR,l[f; c]. Section 3 deals with the sharpness of the lower bound N,,l(f; concerning

the sharpness of N(f;

c). The existing results

c) for mappings between closed oriented n-manifolds,

n # 2, are extended to the setting of maps of pairs, not only when X and Y are closed manifolds,

but also when X and Y are manifolds

preserving

map. As suggested by Example

with boundary

case depend on whether or not c E aY. If c $ aY, the situation for closed manifolds, but the case c E aY is quite different. Section 4 is concerned

with extending

and f is a boundary-

1.1, the results in the manifold with boundary is very similar to that

to maps of pairs an important

computational

result due to Hopf [12, Satz V and Satz VIIa] and Brooks [2, Corollary 2, p. 7251. Given a map f : X + Y and a point c E Y, if c $! f(X) of course MR[p; c] = 0 so N(f;c) = 0. Otherwise, choose 50 E f-‘(c), let F denote the image of the induced homomorphism fn : 7ri (X, ~0) --+ ~1 (Y, c) and let rri (Y, c)/.F be the set of cosets. Theorem

1.2 (Hopf [12], Brooks [2]). If f : X + Y is a map where Y is a closed topological manifold, and c E Y is any point, then either all the root classes off at c are inessential or all are essential, and therefore either N(f;

c) =: 0

or

N(f;

c) = #(7rl (Y, c)/F).

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In Section 4 we will prove extensions

of this result to maps f : (X, A) -+ (Y, B),

both for the case c E B and for the case c $ B. We will see that in addition the homomorphisms inclusions

of subspaces

Coincidence developed,

of fundamental

groups induced by f, homomorphisms

into Y play an important

role in these results.

for maps of pairs f, g : (X, A) -+ (Y, B) have recently

theories

independently,

by Jezierski

to

induced by been

[ 151, and by Jang and Lee [ 131. However, their

theories do not specialize, in the case that g is the constant map, to the approach to relative root theory presented here. Nor do they imply the results of this paper, even if A and B satisfy the restrictive

assumption,

connected. In particular, Theorem are false for coincidences.

made in both [15] and [13], that they be

1.2 and its extensions

to maps of pairs in Section 4

A Nielsen number for roots of maps of pairs f : (X, A) + (Y, B) has been proposed by Yang [26] under the assumption that c E B. Yang showed that his number is a homotopy invariant lower bound, but did not discuss the sharpness of his number. We will prove, in Theorems 3.3 and 3.4(ii) below, that Yang’s number is a sharp lower bound for maps between closed oriented n-manifolds, n # 2. But in a more general setting his number gives less precise information

than the one we use, as we will show in Example 2.4.

A very readable exposition of many results of Nielsen root theory, based on Brooks’ work, can be found in Kiang’s book [ 161, and we shall refer to it whenever possible. In particular, we shall follow Kiang and characterize the essentiality of a root class by its homotopy invariance rather than by its nonzero multiplicity, and thus use the quite general definition of N( f ; c) due to Brooks rather than the one introduced in the manifold setting by Hopf. These definitions are equivalent for maps between closed oriented manifolds of the same dimension

(see Remark 3.2).

2. The relative root Nielsen number Let X and Y be topological spaces and let A and B be closed locally path-connected subspaces of X and Y, respectively. We shall assume throughout this paper that X and Y are compact, path-connected

and locally path-connected

Hausdorff

spaces, but we do

of the subspaces A and B. For f : (X, A) -+ (Y, B) a + B the restriction of f to A. Let c E Y be any point. map, we will denote by f : A We recall from [ 16, p. 1241 that the root classes of f : X + Y are the equivalence classes of root(f; c) under the following equivalence relation. Points 2,~’ E root(f; c) not assume path-connectedness

are equivalent if there is a path < in X from z to x’ such that [f(C)] = 1 E ~1 (Y, c). The definition of root class applies as well to the restriction f : A + B, keeping in mind that there are no root classes if c +! B and that if x,x’ E A are not in the same path component, they cannot be members of the same root class. Each root class of f is contained in some root class of f : X + Y. We will also apply the root class concept to ahomotopyK:Xx1AY. In order to define N,l(f; c), we will need to impose conditions so that there are only a finite number of root classes. If c E B, let B’ denote the path-component of B that

R.R Brown, H. Schirmer / Topology and its Applications

contains

c and denote by A* the union of the path-components

to B*. According

to our assumption,

251

92 0999) 247-274

of A that are mapped

X and A* are compact.

If, further, X and A*

are locally path-connected

and if Y and B* are locally simply-connected, then by [16, ---f Y and J: A 4 B have only finitely Chapter V, Theorem 3.4, p. 1261 the maps f : X many root classes at c. For a set S c X x I and t E I, define the t-slice [S], of S to be

[S]t = {LcE

x:

(x,t)

E

s}.

Suppose K: X x I + Y is a homotopy,

so we have the maps kt :X + Y defined by

kt(z) = K(z, t). G iven a root class R of ku at c E Y, there is a root class Iw of K at c containing R, and each nonempty t-slice [RI, is a root class of kt so, in particular, [R]e = R (see Lemmas 4.1 and 4.2 of [7], in the case that g is the constant map at c). A root class R of f : X -+ Y is inessential if there is a homotopy K: X x I --) Y such that ka = f with [lR]s = R and [IR]r = 0. A root class that is not inessential

is

said to be essential and the Nielsen number N(f; c) of roots of f at c is defined to be the number of essential root classes for f at c. Since the definition of essentiality of root classes did not require that the spaces be path-connected, for a map of pairs f:(X,A) ---$ (Y,B) we also have the Nielsen number of the restricted map f : A --+ B, namely, N(J, c) is the number of essential root classes of f at c. If AT, . . . , Ah are the components of A* and we let & : A5 + B” be the restrictions of f to these components, then

N(J;c)= eN(f,;c), j=1

In the setting of maps of pairs, we need to introduce essential

a modification

of the concept of

root class, as follows.

Definition 2.1. Let f : (X, A) --+ (Y, B) be a map and c E Y a point. A root class R of f : X + Y at c is relatively inessential if there is a homotopy of pairs K : (X x I, AxI)

+

(Y,B)

such that ku =

property [IR]r = 0. Otherwise, will use the notation of f at c.

N+(f;

f

and the root class IR of K with [Ii%]0= R has the

we say that the root class R of

f

is relatively essential. We

c) to denote the number of relatively

essential

root classes

A relatively inessential root class is inessential, so N+(f; c) > N(f; c). For the map f: (D, aD) 4 (0, aD) of E xample 1.l and any c E D, there is a single root class for f : D + D. It is inessential, thus N(f; c) = 0, but it is relatively essential so N+(f;c) = 1. If, for a map f : (X, A) 4 (Y, B) and c E Y, a root class R of f : X + Y at c contains an essential root class ?? of f : A --f B at c, then R is called a common root class of f and f at c. We denote the number of relatively essential common root classes

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RX Brown, H. Schimer / Topology and its Applications 92 (1999) 247-274

by N+ (f, f; c). We can now define the principal pairs.

tool for the study of roots of maps of

Definition 2.2. Let f : (X, A) + (Y, B) be a map and choose c E Y. Suppose X and A* are locally path-connected

spaces and both Y and B’ are locally simply-connected.

Define iV,,l(f; c), the relative Nielsen number of roots of f : (X, A) +

(Y, B) at c, as

follows:

N&f; c) = N(f; c) + Nf(f;

c) - Nf(f,

f;

(g.

It follows easily from the definition that N&f; c) 3 N(f; c). We also note that N,,l(f;c) = N(f;c) if B = 0, so the relative Nielsen number specializes to the root Nielsen number when f is not a map of pairs. Although the definition of N,,l(f; c) applies to any point c E Y, if c $ B there are no root classes of f, so the definition simplifies to N,l(f; c) = N+(f; c). For the map of Example 1.1, we have N,l( f ; c) = Id] if c E B = dD and N,l( f ; c) = 1 otherwise, illustrates

so Nrel(f; c) = MR,Jp;

the definition

c] for this example. The following

simple example

when the spaces A and B are disconnected.

Example 2.3. Let X = Y be the annulus 5” x I and let A = B be its boundary. We suppose that f : (X, A) -+ (Y, B) is a map of degree d # 0. If B = f(A), then N&; c) = N(f; c) if c E B and N,,l(f; c) = N+(f; c) otherwise, and N&f; c) = IdI in either case. If B # f(A), that is, f maps both components of A to one component of B, then Nrel(f; c) = N(f; c) = 2jdl if c E f(A) and N&; c) = 0 otherwise. we have in all cases N,,,(f; c) = MR&; c] whereas N(f; c) = 0. Not only may the value of N,,l(f;

c) depend on whether or not c is in

Again

B, as it does

in Examples 1.1 and 2.3, it may even depend on the location of c within B, as we will demonstrate in Example 3.15. The definition

for the relative Nielsen number

of roots proposed by K.-Y. Yang [26,

Definition 2.31 for the case c E B is N(f; c) + N(f; c) - N(f, f; c) where N(f, j; c) denotes the number of essential root classes of f :X + Y that contain essential root classes of f. In other words, the definition

is the same as N&f;

c) but without

the

concept of relative essentiality of root classes. Since an essential root class is relatively essential, we see that N(f; c) - N(f, f, c) < N+(f; c) - N+(f, f, c). The following example demonstrates that N&f; c) can be a sharper estimate of MR,Jf; c] than the number in [26].

Example 2.4. Let X = Y be the unit disc in the plane and let A = B be the union of the unit circle and the segment of points (t,O) such that 0 < t < 1, and let c = (0,O). We view X as the cone over (0,O) and Y as the cone over (l/2,0), and we define a map f : (X, A) + (Y, B) by sending (0,O) to (l/2,0) and extending by coning. Since c is not in the image of f, the definition from [26] reduces to N(f; c) which equals 0. On the other hand, NT&; c) = N+(f; c) = 1 = M&[f; c].

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R.F: Brown, H. Schirmer / Topology and its Applications 92 (1999) 247-274

Theorem 2.5. Let f : (X, A) +

(Y, B)

and A* are locally path-connected Then f: (X,A) homotopic

+

(Y,B)

to f : (X, A) +

Consequently,

NT&;

be a map and choose c E Y. Suppose

that X

and that Y and B* are locally simply-connected.

h as at least NTPl(f; c) roots at c. If g : (X, A) --t (Y. B) is (Y, B) by a homotopy of pczirs, then r\‘,,l(f; c) = N,,l(g;

c).

c) < MR,,~[f; c].

Proof. The lower bound property of N,,l(f;

c ) can be proved by a straightforward

adap-

tation of the one given for the relative Nielsen number in [21, Theorem 3.11. To obtain homotopy invariance of N&f; c), it is necessary to show that N’(f; c) and N+(f, f; c) are invariant under a homotopy of pairs. For N+(f; c), we can parallel the proof of the invariance of N(f; c) under a homotopy H : X x I -+ Y (see, e.g., [ 16, Chapter V, Theorem 4.4, p. 1291). The proof that N+(f, f; c ) is invariant under a homotopy of pairs can easily be done by an adaptation details. 0

of the proof of 121, Theorem

3.31. We omit the

3. Sharpness of N,,l(f; c) The root Nielsen number N(f; c) of a map f : X ---f Y is called sharp if it is a sharp lower bound for the number of roots for all maps in the homotopy class of f, that is, if N(f; c) = MR[f; c]. Sharpness of N(f; c) 1s established by constructing, in the proof of a “minimum theorem”, a map g N f with p recisely ,‘V(f; c) roots at c. In general this can be done only in a manifold setting. It is known that the root Nielsen number N(f; c) of f : X -+ Y is sharp if both X and Y are closed, connected, oriented PL n-manifolds and n # 2. (See [12, Satz XIIIb] and [18, Theorem B] for n 3 3; n = 1 is trivial, and it is shown in [ 12, Satz XV, XVa and XVb] and [18, $41 that there exist maps from the double torus to the torus for which N(f;

c) is not sharp.) Further, N(f;

c) is sharp if X

is path-connected and Y a topological manifold (compact or not) which is not closed [6, Theorem 7.41. In this case N(f; c) = 0 for all maps f : X + Y and c E Y. We extend root Nielsen

the definition number

N,,l(f;

of sharpness

to maps of pairs, and say that the relative

c) of a map f : (X, A) -+ (Y, B) is sharp if N,,l(f;

c) =

MR,,l[f; c]. Again we will be able to establish sharpness of NTel(f; c) only in manifold settings. More precisely, we will deal with three cases which correspond to cases that have been studied in nonrelative root and coincidence theory (see, e.g., [12,18] and 171) and in relative coincidence theory (see [15]). In all cases, we assume that X and Y are connected oriented manifolds of the same dimension. In the first case, Theorem 3.3, we will consider manifolds X and Y which are closed. In the second and third case we will deal with manifolds X and Y that have a nonempty boundary, and then assume that f: (X,A) + (Y,B)is . a b oundary-preserving map (i.e., f(aX) c aY). We will see that in the first case the location of c in Y is not important. If X and Y have a nonempty boundary, then the results in the second case, where we assume c E int Y, resemble those in the first case, but they are quite distinct from those in the third case, where we assume c E aY. (See Theorems 3.9 and 3.13.) In the second case our definition of N,,l(f;c)

254

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Brown, H. Schirmer / Topology and its Applications

92 (1999) 247-274

will be needed to obtain a sharp lower bound. But in the first and third case we will show that simpler Nielsen

type numbers,

N,l(f; c), are equal to N,l(f; and 3.13.)

which in general

are only lower bounds

c) in these cases and hence sharp. (See Theorems

Note that, according to the assumptions ifolds are compact and connected,

made in Section 2, all manifolds

although

the compactness

assumptions

for

3.3,3.4

and submancan often be

replaced by compact-type but weaker conditions on root(f; c) (compare, e.g., [12] and [15]). As we want to use results from [18], we assume from now on that all manifolds are PL manifolds, but our proofs can be adapted to hold for smooth manifolds (see [ 181) and (using ideas from [15] and [14]) for topological manifolds. We will now establish sharpness of N,,l(f; c) by proving a minimum theorem, Theorem 3.3, in our first case where X and Y are closed oriented n-manifolds. It is clear that N,,l(f; c) can only be a sharp lower bound for the number of roots of f : (X, A) + (Y, B) if N(f; c) is a sharp lower bound for the number of roots of f, and so it is necessary to include sharpness of N(f; c ) as an assumption. Since A* is not necessarily pathconnected,

sharpness of N(f; c) in the disconnected case means that all N(fj; c) are of A*. We sharp, where fj : Aj 4 B* denotes the restriction of f to a path-component shall further have to assume that A can be by-passed in X, which is an assumption

frequently needed in relative Nielsen theory (see, e.g., [21,15]). But it is not necessary to assume that B can be by-passed in Y. We will also assume in Theorem 3.3 that all components of A are closed submanifolds of X. This assumption could be relaxed and submanifolds with boundary could be included. But the proof becomes longer and more complex, as some constructions we will quote from the proof of Theorem 2.4 in [I51 would have to be adjusted. In the later minimum theorems, which concern manifolds with boundary, we could follow [15] and additionally include closed submanifolds of the boundary of X and Y in (more complicated) hypotheses on A and B, respectively. As we do not have any compelling examples to motivate them, we have omitted these somewhat cumbersome generalizations. In the proof of Theorem 3.3, we will use the multiplicity m(R) of a root class R of a map f : X + Y between closed oriented manifolds

of the same dimension

defined by

m(R) = deg,W). Here U is an open subset of X containing R but not containing any roots of f which do not lie in R, and deg,(flU) is the local degree of f over U. (See, e.g., [9, p. 2671 for a definition of the local degree. See [ 181 for the definition of the multiplicity, as well as [ 12, Definition VIIa], see also the sketch of an equivalent definition of the multiplicity of such a root class, as well as of an index of a root class in a much more general setting, in [16, Chapter V.7, p. 136-1381 which is based on [2].) The multiplicity of a root class is homotopy invariant [ 18, Proposition 31. It follows immediately from the definition of m(R) and the additivity of the local degree (see, e.g., [9, Proposition 4.7, p. 2691) that

de&f) =

CdR),

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RX Brown, H. Schirmer / Topology and its Applications 92 (1999) 247-274

where the summation is taken over all root classes of f at c. The degree deg(f) of f can be used to make Theorem 1.2 more precise when the domain is a manifold, as follows.

Theorem 3.1. Let f : X -+ Y be a all root classes off

N(f;c) =

map between two oriented

closed n-manifolds.

Then

have the same multiplicity, and

#(XI(K c)/JY, ifdeg(f) # 0, o

if deg(f) = 0.

>

Proof. For deg(f)

# 0, this theorem is due to Hopf [12, Satz V and VIIa] and to Brooks [2], see also [16, Corollary 7.3, p. 1381. If deg(f) = 0, then it follows from 112, Satz XIVb

and XIVc] or from [18, Theorem A] that MR[f; c] = 0, and so 0 6 N(f;

c) < MR[f; c]

shows that N(f; c) = 0. This implies that all root classes are inessential, have the same multiplicity m(R) = 0. 0

and hence all

Remark 3.2. It is obvious from the definition variance

that an inessential

of the multiplicity and its homotopy inroot class has zero multiplicity. On the other hand, it is not

obvious

from the definition

that an essential

root class has nonzero

multiplicity.

But it

follows from Theorem 3.1 and [ 18, Proposition 51 that, for a map between oriented closed n-manifolds, the root Nielsen number used by Hopf in [ 121 and Lin in [18], which is defined as the number of root classes with nonzero multiplicity, equals the root Nielsen number N(f; c ) used in this paper, namely the one introduced by Brooks in [l] and defined as the number of root classes which cannot be removed under a homotopy. root class has nonzero multiplicity if and only if it essential.

Theorem 3.3 (Minimum ented n-manifolds, submanifolds

is sharp, then N,,l(f;

ori-

where n # 2, and let c E Y. Let A be a disjoint union of closed of Y. Assume, furthen

(Y, B) is a map of pairs so that N(J;

c)

c) is sharp.

Proof. We have to construct

a map g z

c) roots at c. The construction

the manifold

and Y be closed

of X, and let B be a disjoint union of submanifolds

that A can be by-passed in X. If f : (X, A) +

N,,(f;

Let X

theorem for closed manifolds).

So a

f : (X, A)

is essentially

setting, of the proof of the Minimum

4

(Y, B)

an adaptation, Theorem

which has precisely and simplification

to

6.2 in [21]. (See also the

proof of Theorem 2.4 in [ 151.) If n = 1, men X = Y = S’ and A = 0, so the theorem is obviously

true in this

case, and we can assume that R. > 3. If c E B, we first use the assumption that N(f; c) is sharp to homotope 7: A + B to a map with N(f, c) roots which, to simplify the notation, we continue to call f. If c $ B, then N(f; c) = 0 and f : A ---f B has no root at c, so f already has the required property. The homotopy extension theorem allows us to extend f to all of X. The resulting map can be deformed near each one of the finitely many points of root(J; c), with no change to the map on A, to a map with the property that its set of roots on X - A is closed in X. Then we can use transversality to deform the map in a neighborhood of the set of its roots on X -- A, and relative to A, to obtain a

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map which is related to f : (X, A) + (Y, B) by a homotopy

of pairs, has N(F, c) roots

in A and finitely many roots on X - A. If R is any root class of this map which has more than one point on R n (X - A), we proceed as in [ 18, $3, proof of Theorem B] and unite these roots to a single root. Thus we construct a map which has in any root class at most one root on X - A, and this root has nonzero multiplicity.

As A can be by-passed

from [ 181 can be carried out in X - A. Thus we obtain a map

in X, all constructions

which is related to f : (X, A) 4

(Y, B) by a h omotopy of pairs, has N(f; c) roots on A, has finitely many roots on X - A which are all of nonzero multiplicity, and has the property that no two roots on X - A belong to the same root class. We call this map f’ : (X, A) 4 (Y, B). Now let 2 E X - A and a E A be roots which belong to the same root class R of f’ (the root class is common in the sense of Section 2 because f has N(f; c) roots). Then it is necessary to construct a map f” M f’ : (X, A) + (Y, II) so that its set of roots is root(f”,c) = root(f’,c) - {cE). Th is can be done by using the construction which is carried out in the proof of Theorem 2.4 of [15] on the map g in the case where x0 E int M, but replacing the neighborhood V by a euclidean neighborhood of c, the arc 7 by a closed curve at c, and the map f and its changes under homotopies constant map at c. We leave it to the reader to check the details.

by the

After carrying out this construction for all such roots 5 and a, we obtain a map that has on X - A only isolated roots, and each of those roots represents a distinct root class of f : X -+ Y which does not contain a root class of f and has nonzero multiplicity. As such a root class is essential, map g z f with N,,l(f;

it must be relatively

c) roots on X.

essential.

Thus we have obtained

a

0

Next we will show, in Theorem 3.4(ii), that N&f; c) can be simplified if the assumptions of Theorem 3.3 hold, and that it equals in this case the relative root Nielsen number proposed in [26, Definition 2.31, where essential rather than relatively essential root classes of f : X -+ Y are used. Therefore number is sharp under the assumptions

it follows from Theorem

3.3 that Yang’s

of this theorem.

Theorem 3.4. Let f : (X, A) -+ (Y, B) b e a map of pairs which satisfies the assumptions of Theorem 3.3. (i) A noncommon root class R off : X + Y is relatively essential if and only if it is essential. (ii) N,.&;c) = N(f;,c) + N(f;c) - N(f, f;c), where N(f,f;c) is the number of essential root classes of f : X --f Y which contain an essential root class of ~:A-,B.

Proof. (i> An essential root class is relatively essential, so it suffices to show that a root class R of f : X +

Y which is inessential

and not common

is relatively

inessential.

Since R is not common we can, as in the proof of Theorem 3.3, construct a map f’ M f so that all roots on A lie in essential root classes of f”, and so that f’ : X + Y either has no root class corresponding to R or has a root class R’ corresponding to R which

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consists of a single point 2’ E X - A. As R is an inessential class R’ = {cI?}. Therefore neighborhood

m(R)

root class, so is the root

= 0, and we can remove Z’ by a local change in a

CJ c X - A of 2’. The homotopy used in this removal is relative to U,

and so it is a homotopy

of pairs. Hence R’, and thus R, is a relatively

inessential

root

class. (ii) We have to prove that

Nf(f; c) - N+((f,f; c) =

N(f;

C) - N(f,

f;

(,),

i.e., that a root class of f : X + Y which is not common is relatively only if it is essential. But this follows immediately from (i). •I

essential

if and

Example 3.5. Let X = Y = S’ x S’ x S’, let A = B be a circle contractible

in X = Y and take c E B. Then the hypotheses of Theorem 3.3 are satisfied. Let f : (X, A) + (Y, B) be a map where f : X 4 Y and f : A -+ B are both of nonzero degree and note that, since B is contractible in Y, all the root classes of J are in a single root class R

of f. Then N(f; c) = 1deg(f)l by [4, Theorem 1, p. 4071, so f : X + Y has 1deg(f)l essential root classes, of which one, namely R, is common. The same result implies that N(f; c) = 1deg(f) 1. By Theorem 3.4(i), the noncommon only if they are relatively essential. Therefore N+(f; because

c) - N+(f,

fi c) = I deg(f)l

if R is not relatively

essential,

root classes are essential if and

- I it is not counted

in N+(f;c)

and, if it is,

N+ (f, f; c) = 1. Thus, applying Theorem 3.3, there is a map g M f with exactly N&; c) = I deg(f)] + ) deg(f)j - 1 roots at c. In the remainder of this section, we shall consider maps between oriented n-manifolds with nonempty boundaries and, in Theorems 3.9 and 3.13 below, we will extend the Minimum

Theorem

3.3 to this setting. We will always assume that the map f : X + Y

is a boundary-preserving map, which means that f maps 8X to aY. We will also assume that dX c A and aY c B and that all maps of pairs f : (X, A) + (Y, B) have the additional property of being boundary-preserving. Further, we will assume that all homotopies of pairs H : (X x I, A x I) -+ (Y, B) are boundary-preserving, that is, H(BA x I) c aB. We will call such a homotopy a boundary-preserving homotopy of pairs. The proof of the Minimum Theorem 3.3 made use of the concept of the multiplicity of a root class for a map f : X + Y between closed oriented manifolds. A multiplicity is also needed to prove a minimum theorem in the case of a boundary-preserving map between manifolds with nonempty next define such a multiplicity.

boundaries

if we assume c E int Y. Therefore we will

So let X, Y be oriented n-manifolds with boundary, let f be a boundary-preserving map and let c E int Y. In this case we use the doubles 2X = X+ U X- and 2Y = Y+ U Y_ of X and Y, and identify X with X+ and Y with Y+. As the map f is boundary-preserving, it defines the double 2f : 2X + 2Y and, as c E int Y, the set of

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roots of 2f at c lies in X+ and can be identified with the set of roots of f at c. Moreover, the Nielsen equivalence

in these two root sets is the same due to

Lemma 3.6. Let f : (X, ax)

-+ (Y, aY) b e a map between two manifolds with boundary

and let c E int Y. Then the root classes of root(2f,

c) are identical to those of root(f, c).

Proof. A proof can be obtained by replacing f by the constant map of X to c and g by f in the proof of Lemma 5.1 in [7].

0

We use Lemma 3.6 to define the multiplicity

m(R)

of a root class R of f : (X, ax)

of R considered as a root class R c X+ of 2f : 2X + 2Y. w h ere R c U c 2X for an open set U in 2X which Hence m(R) = deg,(2f[U), contains no roots of 2f which do not lie in R. Note that c E int Y implies R C int X, and therefore the set U used in the definition of m(R) can be chosen as a subset of 4

(Y, aY) as the multiplicity

int X = int X+. This fact will be important when we prove the next minimum theorem, Theorem 3.9, as all constructions needed to minimize the set of roots will have to be carried out in X = X+. The homotopy invariance

of the multiplicity

of a root class is stated in the next theorem.

Theorem 3.7. Let fo, fi : (X,3X) two oriented n-manifolds

maps between -+ (Y, 3Y) be b oundary-preserving with boundary, let H : (X x I, aX x I) -+ (Y, aY) be a boun-

dary-preserving homotopy from fo to fl, and let c E int Y. Zf the root class & of fo at c corresponds to the root class RI of fl at c under this homotopy, then m(Ro) = m(R1).

Proof. This theorem is an immediate

consequence

of the homotopy

invariance

of the

multiplicity of a root class of the map 2f, for the homotopy H defines a homotopy 2H : 2X + 2Y, and I& and RI are root classes of 2f0 and 2fl which correspond under this homotopy.

0

i s a boundary-preserving map between oriented n-maniIff: (x,ax)+ (y,ay) folds with boundary, then deg(2f) = deg(f), w h ere the degree deg(f) of f is defined, as usual, by means of the induced homomorphisms fn* : H,(X, ax) -+ H,(Y, au). Hence it follows from the additivity

hdf) =

of the local degree that if c E int Y, then

Cm(R),

where the summation is taken over all root classes of f at c. The following lemma relates relatively essential root classes of boundary-preserving maps to those of nonzero multiplicity. Note that a boundary-preserving map is a special case of a map of a pair, and so “relatively essential” in the statement of Lemma 3.8 means essential with respect to homotopies of the form H : (X x I, aX x I) + (Y, au).

Lemma 3.8. Let R be a root class of a boundary-preserving between two oriented n-manifolds R is relatively essential.

map f : (X, ax) + (Y, aY) with bou.-zdary, and let c E int Y. If m(R) # 0, then

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Proof. Assume, by way of contradiction,

that R is a relatively

f. This means that there exists a homotopy

inessential

R corresponds

to the empty set. The double 2H defines a homotopy

R, considered

as a root class of 2 f, corresponds

root class of 2f. But then m(R)

root class of

H : (X x 1. aX x I) ---f (Y, aY) under which of 2f under which

to the empty set, so R is an inessential 0 the assumption.

= 0, which contradicts

We are now ready to prove a minimum case c E intY.

theorem for boundary-preserving

maps for the

Theorem 3.9 (Minimum theorem for manifolds with boundary and c E int Y). Let X and Y be oriented n-manifolds with boundary, where n # 2, and let c E int Y. Let A be the disjoint union of aX and closed submanifolds

of X, and let B be the disjoint

union of 3Y and submanifolds of Y. Assume, further; that A can be by-passed in X. If and if N(f; c) is f : (X. 4 4 (Y B) is a map of pairs which is boundary-preserving sharp, then N,,l(f;

c) is sharp.

Proof. As it is again very easy to see that the theorem is true if 7~= 1, we assume that n 3 3. Using the assumption

that N(f,

c) is sharp, as well as homotopy

extension

and

transversality, we can (as in the proof of Theorem 3.3) homotope f : (X, A) + (Y, B) relative to A to a boundary-preserving map f’ : (X, A) + (Y, B) which has N(J, c) roots on A and finitely many roots on X - A. This map defines a map 2f’: (2X, 2A) + (2Y, 2B) and, according to Lemma 3.6 the map 2f’ has the same roots and root classes as f’. Since all roots of 2f’ lie in int X+, we can proceed as in the proof of Theorem 3.3, and make sure that all changes to 27 are carried out in int X+. The resulting homotopy of 2f’, which is relative to X_, yields by restriction to X+ a map g with the property that each remaining root on X - A represents a noncommon root class with nonzero multiplicity. It follows from Lemma 3.8 that such a root represents a root class which is relatively essential with respect to homotopies of the form H: (X x I, dX x I) + (Y, aY). As 3X C A, such a root class is relatively essential with respect to boundary-preserving 0 homotopies of the form H : (X x I, Ax I) + (Y, B). Therefore g is the required map. Example

3.10. Let X = Y be the n-ball

disjoint embedded

for n 3 3. Let A be the union of 3X and

circles C, in int X for j = 1, . . , m. Let B = ZIY U B*, where B* is

a circle embedded in int Y, and of course c E B*. We observe that A can be by-passed. Consider any maps af: aX + aY and f, : C, + B’, where the degree of f,, with respect to the chosen orientations of the Cj and B’, is d3, and at least one dj # 0. Note that N(f,; c) is sharp for all j. Since Y is an absolute retract, we can extend the maps af and f, to a map f : (X, A) --f (Y, B). S’mce Y is simply-connected, there is only one root class of f : X + Y and, since it contains of these is essential, we conclude that

N&f; c) = N(.f;c) =

c

all the root classes of J and at least one

ldjl.

.,=I

Theorem

3.9 then tells us that there is a map g M .f with exactly Cz”=, Id?\ roots at c.

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R.E Brown, H. Schirmer / Topology and ifs Applicafions 92 (1999) 247-274

We will employ below, an extension

ideas from the proof of Theorem of Theorem

3.4 to the manifold

if c E int Y. Hence the simpler concept relatively consider

3.9 to obtain,

in Theorem

with nonempty

of an essential

boundary

3.11 setting

root class can replace that of

essential root class if the double is used. In the statement of Theorem 3.11, we A as a subset of X+ and identify

the map f : A ---)B with the restriction

2f : 2X ---) 2Y to A c X+. Note that Remark 3.2 implies that, in Theorem multiplicity

m(R)

of

3.1 l(i), the

is nonzero if and only if R is an essential root class of 2f.

Theorem 3.11. Let f : (X, A) 3

(Y, B) b e a map of pairs which satisjes the assump-

tions of Theorem 3.9. (i) A noncommon root class R of f :X

+

Y is relatively essential if and only if

m(R) # 0, (ii) N,l(f; C) = N(f; c) + N(2f; c) - N(2f, f; c), where N(2f, f; c) is the number of essential root classes of 2f : 2X -+ 2Y which contain an essential root class of f:Ai B. Proof. (i) A root class which is relatively essential with respect to homotopies of the form + (Y, aY) is relatively essential with respect to boundary-preserving H: (Xx1,axxr) homotopies

of the form H : (X x 1, A x I) -+ (Y, B). Therefore, due to Lemma 3.8, it is

only necessary to prove that if R is a noncommon root class with m(R) = 0, then R is relatively inessential. As the assumptions of Theorem 3.9 are satisfied, we can proceed as in the proof of Theorem 3.9 and construct a boundary-preserving homotopy of pairs which unites the points in R to a single root 20 E X - A of a boundary-preserving map. Let the resulting map be f” : (X, A) + (Y, B). According to Theorem 3.7, the multiplicity m({zc>) = 0. It remains to show that (~0) can be removed by a boundatypreserving homotopy of pairs. To do so, we identify X with X+ and A with A+ in the double 2X. The isolated root x0 has a euclidean {x0}. As ~0 E X, a homotopy

neighborhood U in 2X which contains no roots of 2f” other than - A+, we can choose U c X+ - A+ c int X+. Hence there exists

of 2 f” which removes ~0 from the root set of 2 f” by a local change on U.

This homotopy

is a homotopy

boundary-preserving

homotopy

relative to 2X - U, so it defines by restriction H : (X, A) + (Y, B) of

f”

Part (ii) follows from part (i) as in the proof of Theorem

We may now extend Theorem 3.1 to boundary-preserving and write 03 for the image of the homomorphism (2f )x : Tl

to X+ a

which removes R. 3.4(ii).

0

maps. Choose ~0 E f-i(c)

(2X, x0) + Tl (W cl

induced by the double 2f of f. We shall see, in Corollary 4.8 below, that the expression for NTpl(f; c) given in Theorem 3.12 can be put into a form which avoids the double and is therefore easier to use in computations.

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Theorem 3.12. Let f : (X, CIX) + (Y, aY) b e a boundary-preserving oriented n-manifolds

and let c E int Y. Then all root classes off

map between two

have the same multi-

plicity, and

ifded.0 # 0, ifdeg(f) Proof. For deg(f)

# 0, this theorem

= 0.

is an immediate

consequence

of Theorem

3.1,

Lemmas 3.6, 3.8 and our definition of m(R). Now let us assume that deg(f) = 0. If n = 1, the theorem is trivial. If R = 2, we use [8] to construct a map fi M f with = dX. Th’is map f+ is proper in the sense of [24, p. 4161, so we can use

(f+)-‘(au)

the definitions of the algebraic and the geometric degree in [24, p. 4161 to conclude from [24, Theorem 2.41 that there exists a map g z f+ : (X, ax) ---) (Y, dY) with c $ g(X). Hence NTrl(f;c)

= 0. It follows from Theorem

3.1 and Lemma 3.6 that m(R)

= 0 for

any root class R of f at c so, if n 3 3, then Theorem 3.1 l(i) implies that R is relatively inessential. Therefore N,,l(f; c) = 0 in this case as well. 0 Applying Theorem 3.12 to Example 1.1 in the case c = 0 shows us that N,,l( f; c) = 1. Since we proved in Example 1.1 that MR,t[f; c] = 1, we see that the Nielsen number is sharp in this case. Finally

we will consider

the case where f is a boundary-preserving

map between

oriented n-manifolds with boundary and c E dY. In this case no multiplicity theory is needed to prove the equivalent of Theorem 3.9. We can also weaken the assumptions considerably;

in particular,

Theorem 3.13 (Minimum

it is not necessary

theorem for manifolds

and Y be oriented n-manifolds of aX and submanifolds

c) is sharp, then Nr,t(f;

Proof. Using the assumption transversality,

with boundary

and c E dY).

in X. Let

X

with boundary and let c E aY. Let A be the disjoint union

of X, and let B be the disjoint union of i3Y and submanifolds

of Y. Zf f : (X. A) -+ (Y, B) N(F,

to assume that A can be by-passed

is a map of pairs which is boundary-preserving c) = N(f;

that N(f,

we can again homotope

c) and IV,&;

c) is sharp, as well as homotopy f : (X, A) +

and if

c) is sharp.

(Y, B)

relative

extension

and

to A to a map

f’ : (X, A) + (Y, B) which has N(fi c ) roots on A and finitely many roots on X - A. We will show that all of these isolated roots on X - A can be removed by a boundarypreserving homotopy of pairs so that there is a map g = f with N(f; c) roots. Let ~0 E X - A be a root of f’ at c and select a closed euclidean neighborhood U(ZO) of za in X - A that contains no other roots of f’. As in the proof of Theorem 7.2 of [6] for the case a E ax, we can find a compact neighborhood V(c) of c in Y and a map k, : Y -+ Y which maps Y to Y - V(c) and is homotopic to the identity map on Y, and this homotopy can be chosen relative to Y - V(c). As V(c) can be arbitrarily small, we will choose it so that V(c) n f’(W(xo)) = 8. Hence the map k,of’:(X,A) + (Y,B) is related to f’ by a boundary-preserving homotopy of pairs and root(k, o f’, c) = root(f’, c) - (~0). Applying of f’ on X - A gives the desired map g. 0

this removal procedure

to each root

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RX Brown, H. Schirmer / Topology and its Applications

92 (1999) 247-274

Example 3.14. Let X = Y = D the unit disc, and let A be the union of Ca = aD and m disjoint circles Cj in int D. We let B = ao

and choose c E do.

For j = 0, . . . , m,

take any maps & : Cj 4 B, where the degree of fj is dj with respect to the orientation induced

on all circles from an orientation

of the plane. Since D is an absolute retract,

we can extend the map J: A 4 B, determined by the fj, to a map f : (X, A) --+ (Y, B). We note that N(fj; c) is sharp for all j. Thus we may apply Theorem 3.13 to conclude that

Furthermore, by Theorem 3.13 there is a map g x f with x7==, ldjl roots at c. Since each Cj must contain at least ldj / roots at c, then g cannot have any roots at c in X - A. In particular, if we choose the fli so that each has exactly Id31 roots at c, we see from the proof of Theorem 3.13 that we can extend f to the map f : (X, A) -+ (Y, B) without adding roots at c, that is, we can extend f in such a way that the extended map has no roots at c in X - A. We note that in the case m = 0, the setting of this example is that of Example

1.1, with c = 1.

The final example of this section shows that the value of N&f;

c) can depend on the

location of c within B. We note that, although the spaces X and Y are manifolds with boundary, unlike the maps of the preceding section, the map f : (X, A) --+ (Y, B) is not boundary-preserving.

Thus, this example gives us some indication

have avoided by restricting

ourselves

to boundary-preserving

of the pathology

we

maps.

Example 3.15. Let X = Y = St x I and represent plane with l/3

Y as the set of points p in the of X, which we write in terms of

< IpI < 1. Let A be the boundary

its components as A = A0 U Al. Let B = BO U B1 where BO is the circle of points p with Ipl = l/3 and B1 is the annulus of points p such that 2/3 < Ip( < 1. Let f:(X,A)

+

(Y,B)

be a map such that f(Ao)

that the degree d of the restriction

c

BO and f(Al)

of f to A0 is nonzero.

c

BI. We assume

It is clear that N,,l(f;

c) = 0

if ICI > 2/3. We claim that if ICI = 2/3, then N,,l(f;c) = IdI. To verify our claim, the space 2 defined to be the annulus of points p in the plane such that l/3 < IpI < 2/3 and note that c E 32. Let T: (Y, B) -+ (2, a2) be the radial retraction

we consider

and let j : (2, 82) 4 (Y, B) be inclusion, so j o T M id, the identity. Thus f x j o T o f, which implies N&f; c) = N,&’ o T o f; c) by Theorem 2.5. Now we consider the map T o f : (X, i3X) + (2, 32) and relate it to j o T o f. It is clear that these maps have the same roots at c and the root classes are identical because j, : ~(2, c) --) rrt (Y, c) is an isomorphism. If the relative inessentiality of a root class R of j o T o f is demonstrated by a homotopy K : (X x I, A x I) ---) (Y, B) (see Definition 2.2), then the homotopy of pairs TOK establishes the relative inessentiality of R as a root class of TOj o T o f = TO f. Similarly, a relatively inessential root class of T 0 f is relatively inessential as a root class

R.E Brown, H. Schirrner / Topology and its Applications 92 (1999) 247-274

263

of j 0 r 0 f, so we conclude that NT,l(r o f; c) = Nrel(j o T o f; c). The map T o f satisfies the hypotheses of Theorem 3.13, therefore N&U;

c) = N,,l(j

0 r 0 f; c) = N,l(r

0 f; c) = N(r 0 f; c) = IdI

as we claimed.

4. Some calculations of N,,l(f; c) In this section, calculate N,,l(f; homomorphisms

we will present results that allow us, in some interesting c) for a map f : (X, A) + (Y, B) just from information of fundamental groups induced by various maps.

cases, to concerning

Although we restrict our attention to compact spaces X and Y, and to compact subspaces A, A* and B, B*, this assumption can sometimes be omitted completely (e.g., in Lemmas 4.1, 4.4 and 4.6), and can often be omitted for Y and B. We will assume throughout this section that X is path-connected, and that Y is locally path-connected, as well as connected and locally simply-connected as in Section 2. Therefore, Y has covering spaces corresponding to the subgroups of its fundamental group rr’(Y, c), for c E Y. We will use a covering space to replace the definitions of Section 2 by equivalent definitions

that will be more convenient

for presenting

the material of this section. Choose

20 E f-’ (c), which we assume is nonempty since otherwise i’V,l(f; c) = 0, and consider the covering space_pf : $ -+ Y corresponding to the subgroup 3 = fr(rrl (X, ~0)) of 7r’(Y, c). That is, Yf is constructed by defining an equivalence relation on the set of paths in Y that start at c as follows. Paths 0 and r are equivalent if c( 1) = r( 1) and there is a loop w in X based at za such that [a . r-‘1 = [f(w)] E rr’ (Y, c). Denote the class containing 0 by (g) f . The map pf : p’ + Y is given by pf((‘~)f) = o(l). If Q and r are loops at c, then the equivalence condition [g. ~‘1 = [f(w)] can be written as [CJ]= fT( [~])[r], so we see that the fiber pJ’ (c) is the left coset space r’ (Y, c)/F. will call the points of p;‘(c)

We

the factors of f at c.

The space pf is useful in root theory for the following reason. Define a lift f^ : X + ?f off

as follows. For z E X, let pL, be any path in X such that ~~(0) = ~0, the base point

chosen above, and I_L~(1) = 2, then set f(x) definition

is independent

= (f o pZ)f. Once ~0 has been chosen, the

of the choice of the path pZ (see [ 19, Theorem

5.1, p. 1561).

Now, for each 2 E pJ ’ (c), the set f^-’ ( c^) is either empty or a root class of f; moreover, for each root class R there exists cR E ~7’ (c) such that R = f-’ (ER). To be specific, let R be a root class and take some x E R, then f 0~~ is a loop in Y based at c and we have that f^(R) = ?I
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R.E Brown, H. S&inner / Topology and ifs Applications 92 (1999) 247-274

chosen to be a point in A. Write the points of Et, that is equivalence

classes of paths in

B at c, as (.) f. The fiber p7 ’ (c) of gf is the left coset space ~1 (B, c)/F. of pyl (c) the factors of J at c. Define a lift f^’ : A -+ gr

Call the points

of f by p(x)

= (f o j&)f

where ,& is a path in A such that j&(O) = 50 and ,&( 1) = 2. Then a root class z of f at c has the property ?? = cR = 7.

[fo&

= (foj&.)f.

p’- ’ (ER)

w h ere the factor of f corresponding

to R is

W e c h oose as base points of pf and Bf the points & and

cb which correspond to the equivalence class of the constant path at c. Let i : B -+ Y be the inclusion. The map i o pf : Er -+ Y lifts to a map g : gr 4 pf, and we choose i as the (unique)

lift with i(cb) = ho. Clearly i((@)f)

a path in B that starts at c. Restricting pj’ (c) and the restriction

= (i o a)f,

for @

to the factors at c, we see that i maps p?‘(c)

of i is the function

i, : rrr (B, c)/F

-+ 7~ (Y, c)/F

to

induced by

the homomorphism i, : 7rr(B, c) 4 ~1 (Y, c). The computation of N&; c) depends on N+(f, f, c), the number of relatively essential root classes of f that contain essential root classes of _?. Thus the determination of whether a root class of f intersects A is an important part of the root theory of maps of pairs. When A and B are path-connected, the following result characterizes the root classes of f that contain root classes of f. Lemma

4.1. Let f : (X, A) ---) (Y, B) b e a map where A and B are path-connected,

and B are locally path-connected and R be root classes, off ?R = f”(R) E p?‘(c).

and semi-locally simply-connected,

and f, respectively, at c and let ER = f’(x)

Then ??

c

R ifand

only ifi

Y

and c E B. Let ?? E pT1 (c) and

= tR.

Proof. Suppose R c R and choose x E R, then E, = (f o jiL,)f for any path jiL, in A from zc to 5. Since x E R we also have en = (f o ,!&)f. But f o jL:, = i o fo &, so I(?,)

= i((fo

&)j)

= (i 0 fo &)f

= (f 0 j&)f = &.

Conversely, suppose R and R are root classes, of f and f, respectively, at c such that I = CR. We have 6~ = (f o ,!&)f for any path ,!iL, in A from x0 to some x E R. Because I

= tR, we see that

R = f-’

(e,)

= f-’

(f(2,))

= f-’

(; ((f o /!I&))

= f-’

((f o j&).

that x E R and therefore z

Since (f 0 fiz)f = f^(x ) , we conclude

c

R.

0

We will use this lemma to prove our first computational result. It corresponds to a result in the Nielsen theory of coincidences of maps of pairs due to Jang and Lee ([ 13, Theorem 5.61). Theorem

4.2. Let X and Y be closed, connected oriented n-manifolds,

closed, connected

oriented k-manifolds,

a map such that deg(f) N&f;

and deg(f)

c) = #(w (B, c,/F)

let A and B be

k < n, and let c E B. If f : (X, A) +

are nonzero, then

+ #@I (y, c)/F)

- #(i&u

(B, 4/F)).

(Y, B) is

R.F: Brown,

Proof. It follows

H. Schirmer / Topology

from Theorem

all the root classes

of f

and its Applications

265

92 (1999) 247-274

3.1 that all the root classes of f are essential,

are essential

and therefore

relatively

essential,

that

and that

N(f: c) = #(ri (B, c)/F) and N+(f; r) = N(f; c) = #(K, (Y. c)/3),Thus, N+(f, f; c) is the number of root classes of f that contain root classes of f. By Lemma 4.1, a root class Therefore,

II of f

N+(f.

f;c)

contains

a root class z

= #(i@?‘(c)))

of f

if and only

and since z?:pj’(c)

,ix : ~1 (B. c)/F + ~1 (Y, c)/3 induced by the homomorphism q we have proved that N+(f, f; c) = #(i,(r, (B, c)/F)). Example

3.5 satisfies

the hypotheses

=

CR.

is the function

i, : ~1 (B, c) + 7ri (Y: c),

4.2 and [4, Theorem

1, p. 4071

= 1deg(f)j and #(ni (Y, c)/3) = 1deg(f)l. Since, in that in Y, the homomorphism i, is trivial so we again have

implies that #(rr, (B, c),@) example, B is contractible N,,,( f; c) = / de&)

of Theorem

if I

--7‘ p;‘(c)

/ + 1deg(f) I - 1. In the next example, the homomorphism

i, is not

trivial. Example4.3.

LetX=Y=S’xS’,letA=B={l}xS’,choosecEBanddefine

,f : (X, A) ---f (‘Y, B) by f(e16, eie) = (eai9, ebiO) for nonzero integers a and b. Then 7rl(B,c)/T = Z/bZ and ni(Y,c)/3= Z/aZ@Z/bZ. Since i,:Z/bZ --f Z/aZ@Z/bZ is one-to-one,

by Theorem 4.2 we have N,,l(f;c)

= Ibl + labI - lb1 = labi.

Theorem 4.2 requires that A and B be path-connected, so that we can make use of the covering space techniques for f : A + B as well as for f : X + Y. In the rest of this section, we will allow A* to be disconnected and demonstrate may still compute N&f; c) by using other types of hypotheses.

that, in many cases, we The next result indicates

the sort of hypothesis we will require. Let S be a subspace of Y, let c E S, and let i : S --+ Y be inclusion. the subgroup

of “1 (Y, c) that is the image of the homomorphism

x.9 = in(r, (S, c)). F rom [ 19, Proposition

Denote by 1,

induced by i, that is,

11.2, p. 1781 we have

Lemma 4.4. Given a map f : X + Y, the covering space (pf ,pf),

a path-connected

and locally path-connected subspace S of Y and a point c E S, then the subspace pJ ’ (S) of pf is path-connected (f and only if ~1 (Y%c) = 3 .I,. If we choose

xb E f-‘(c),

then th e covering

space pj

corresponding

to 3’

=

fT(rr, (X, z&)) is homeomorphic to Yf by a fiber-preserving homeomorphism, so Lemma 4.4 implies that ?ri (Y, c) = 3 . 1~ if and only if 7rl (Y, c) = 3’ . 1s. Thus the hypothesis TI (Y>c) = 3 ZS is independent of the choice of 1~0E f-’ (c). A natural setting for the hypothesis ~1 (Y, c) = 3.1~ occurs when Y is a Cartesian product Y = Yi x Y2. Writing c = (cl. Q) and letting [~j] denote the class of the constant path in 7ri (Y,. rj), we have T(Y>C)

=

m(X,c1)

=

(m(x,cl)

x

m(y2,c2) x

[c;?]) . ([Cl]

x

Tph)).

266

R.F: Brown, H. Schirmer / Topology and its Applications 92 (1999) 247-274

Thus if, for instance, then the hypothesis

S = c’ x YZ and f : X + Y is such that (7r’(Y’, cl) x [EZ]) c 3, is satisfied.

7r’(Y, c) = 3.2~

Remark 4.5. Let Y be a connected, space, let S be a path-connected f : X -+ Y is any map. Consider

locally path-connected,

subspace

locally

simply-connected

of Y, let c be a point of S and suppose

the double coset space 3 \ r’ (Y, c)/ 1s and, for [S] E

rr’(Y, c), denote the double coset containing [6] by 3. [S] .Zs. For (a) E p?‘(S), choose a path n in S from r~(1) to c. It can be shown that setting r( (0)) = 3. [g. q] .Zs defines a function

r : pJ1 (S) --f 3 \ 7” (Y, c)/ Zs that induces

between the set of path components of p;‘(S) concerns a special case of this correspondence,

a one-to-one

correspondence

and the double coset space. Lemma 4.4 namely, when the double coset space is

trivial. We will define a topological

pair (2, C) to be relatively homogeneous on a subspace

S of Z if, given any points s’ , s2 E S, there is a homotopy of pairs U : (2 x I, C x I) t (2, C) such that ‘ZLO is the identity map and u1 is a homeomorphism with the property U’ (s’) = ~2. It is clear that the subset S must lie entirely in C or entirely in 2 - C. Note that in the next result we do not specify whether c lies in the subspace B.

Lemma 4.6: Given a map f : (X, A) + (Y, B), the covering space (pf,pf)

and a point is relatively homogeneous on the fiber pf I (c), then either all c E Y, if (J$>Fq(B)) the root classes of f at c are relatively essential or all the root classes are relatively inessential.

Proof. Suppose there is a root class R that is relatively inessential.

Let f^ : X -+ pf be a

lift of f : X -+ Y and let I?R E pj ’ (c) be such that R = f -’ (2~). Let R’ be a root class of f at c; we will prove that R’ is also relatively R’ = f-’ (ARC).By hypothesis, (pf,pj’

inessential.

Let 2~ E p; I (c) be such that

there is a homotopy of pairs U : (pf x I,pJ’(B)

(El)) such that us is the identity

map and U’ is a homeomorphism

x I) -+ with the

property U’ (&) = el,. Consider the homotopy of pairs H : (X x 1, A x I) --J (Y, B) defined by H E pf o U o (f x id), where id is the identity map on I, and let H : (X x I, A x I) + (Yf , ~7’ (B)) be the homotopy of pairs given by g = U o (f^ x id). Then 2 ) , so g is the lift of the homotopy H with p,og=Handj?(s,O)=uoof^(z)=f*( iLa = f^. Now let IR’ be the root class of H at c such that its O-slice [?‘I0 = R’. If (z, 0) E [R’]o, then @(x,0) = f(z) = CR!, and so [IV]0 C H-‘(~R/). As H-‘(?R/) is a root class of H, we see that ii-‘(?R,) = w’. But 2~’ is a homeomorphism, so u;’ (ER,) = i‘R and therefore the l-slice of this root class is [R’]’ = f^-’ o U;‘(eR,) = f^-‘(ER). Since f^-I(&) = R is a relatively inessential root class R of f, there is a homotopy of pairs K: (X x I, A x I) + (Y, B) such that ku = f, with [RK] I = 0, where RK is the root class of K for which [RK]o = R. Let g : (X x I, A x I) --) (pf,p;‘(B)) be the lift of K with 60 = f^. (See, e.g., [23, Theorem 2, p. 168 and its proof] for the existence

of a

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R.E Brown, H. Schirmer / Topology and its Applications 92 (1999) 247-274

lifting of a map of pairs.) As [RK]o C E-‘(~R)

homotopy Defining

we see

that RK = I?-‘.

L on X x I by

L(x,t)

=

if 1/2
of pairs L : (X x I, A x I) +

gives us a homotopy consequence

if 0 < t < l/2,

H(z,2t), _ { nfoutoK(z,2t-l),

(Y, B). The continuity

of L is a

of the fact that pf o ut o ,&a= pf o ut 0 f^ = hl. Now letting IRl, be the root

class of L such that [lRl,]a = R’ = ~-‘(CR/), we see that [R’Jt = i;’ o u;‘(cn/) = k;‘(En) = [IRKIt = 0. In other words, the homotopy L demonstrates that R’ is a relatively

inessential

class.

0

We will use Lemma 4.6 to extend Theorem 1.2 in two cases, depending on whether or not c is in B. We have f : (X, A) --+ (Y, B) an d we continue to assume that c E f(X), so we can choose 20 E f-‘(c). As in Theorems 1.2 and 4.2, we will make use of the subgroup F = fr(rt

(X, 20)) of ~1 (Y, c). In the first extension,

c lies outside B.

Theorem 4.7. Let f : (X, A) --f (Y, B) where X and A are locally path-connected spaces, Y is a locally path-connected and semi-locally simply-connected space and Y - B is connected and locally euclidean, and let c E Y - B. If ~1(Y, c) = F . ZY_~, then N&f;

c) = 0

or

N&f;

c) = #(nl (Y, c)/F).

Proof. The subspace p;’ (Y - B) in the covering space $ is locally euclidean because it is locally homeomorphic to Y - B. The hypothesis rrt (Y, c) = F . ZY_B implies by Lemma 4.4 that pJ’ (Y - B) is path-connected. py’(Y-B),takeanarcinpJ’(

Y-B)

arc such that the closure of W is in ~7’ (Y-B). define a continuous

family

Given any two points

(~)f,

that connects them and a neighborhood

(a’)f

E

W of the

As in [25, Lemma 5.4, p. 1321, one may

{ut} of homeomorphism

of $

such that ug is the identity

map, ul((g)f) = (a’)f an d ut is the identity map outside of IV, for all t. Since each ut is the identity on ~7’ (B), we have shown that (Yf , pJ’ (B)) is relatively homogeneous on all of pj’(Y - B) and so, in particular, on the fiber p?‘(c). either all the root classes of f at c are relatively essential

Therefore, by Lemma 4.6, or all the root classes are

relatively inessential. If all root classes are relatively essential and hence nonempty, they are in one-to-one correspondence with the set of factors of f at c. As the set of factors is the coset space rrl(Y, c)/F, relatively

inessential,

The subspace

we have N,l(f;

we clearly have N,,l(f;

B can be by-passed

c) = #(7rr (Y, c)/F). c) = 0.

If all root classes are

0

in the path-connected

space Y if and only if

rr*(Y,c) = z y - B, so Theorem 4.7 can be applied if Y - B is locally euclidean and B can be by-passed. An important instance of this occurs when Y is a manifold with boundary and B is the boundary of Y. Example 1.1, with c the origin, can again be used to illustrate Theorem 4.7 in this case. We next combine Theorem 4.7 with a result from the previous section, to obtain a result concerning homomorphisms of fundamental groups of orientable manifolds induced

268

R.E Brown, H. Schirmer / Topology and its Applications 92 (1999) 247-274

by maps of nonzero degree. We note that the statement of the result makes no explicit mention of roots. Recall that we assume throughout this paper that X and Y are compact and path-connected. Corollary 4.8. Let f : (X, ax) --$ (Y, aY) be a boundary-preserving map between two oriented n-manifolds with boundary such that deg(f) # 0. For c E intY and xo E f-* (c), let F denote the image of fT : ~1(X, $0) -+ 7~1(Y, c) and IV the image of the homomorphism (2f)% : TI (2X, ~0) -+ 7rl(2Y, c) induced by the double off, then #(T* (Y, 4/q

= +rr (2Y, c)lDF).

Proof. Since the degree off is nonzero, Lemma 3.8 implies that all the root classes off are relatively essential, so Theorem 4.7 tells us that NJ f; c) = #(PTI (Y, c)/.F). On the other hand, in Theorem 3.12, deg(f) # 0 implies that N&(f; c) = #(xl (2Y, c)/DF). 0 Corollary 4.8 does not extend to boundary-preserving maps of degree zero. For instance, if X = Y = [0, l] and f(t) = 1 for all 0 < t < 1, then #(rt(Y, c)/F) = 1 whereas #(a-t (2Y, c)/DF) = co. Theorem 4.7 may be applicable when B cannot be by-passed, as the following example demonstrates. Example 4.9. Let X = Y = S’ x 5” and define f : X -+ Y by f (eiB,e’$) = (ekie, e’@‘) for k an odd positive integer. Let i’ : S’ --+ Y be an embedding such that, for ik : Z = 7~(S’) + rl (Y) = Z&Z, we have i:(m) = (2m, m). Defining B to be the complement in Y of an open regular neighborhood of i’(S’) and letting i : Y - B --) Y denote inclusion, i x = ia so, since ik is not onto, B cannot be by-passed in Y. Let c E Y - B and choose some ~0 E f-t(c). We claim that ~1(Y, c) = 3 @ xY_B (where we use additive notation because a-t(Y, c) is abelian). Let (u, U) E Z @ Z = 7rt(Y, c). Recalling that k is odd, we write (U/II)= (,(,,,,(y+=

k(--u),v-

++Fu) “;’ -+

+ (2(q+,qq

(

which verifies the claim. Now let A = f-’ (B), which is locally path-connected because f is a covering map. Then we have shown that, for the map f : (X, A) -+ (Y, B), we have satisfied the hypotheses of Theorem 4.7, so we conclude that N,l( f; c) = 0 or N,r(f;c) is the order of the group 7rt(Y, c)/F = (Z ~3Z)/(kZ 63 Z) = Z/kZ, which is of order k. Since N(f;c) = k in this case [4, Proposition 6.1, p. 4151, we see that N&f; c) = k. Furthermore, f is a k-to-one map, so N,l(f; c) = M&l[f; c], that is, no map of pairs homotopic to f can have fewer than k roots at c. Of course the same is true for any map homotopic to f : X -+ Y because N(f; c) = k.

269

R.E Brown, H. Schirmer / Topology and its Applications 92 (1999) 247-274

We will now extend Theorem 1.2 to the case where c E B. In Theorem 4.7 we required that Y - B be locally euclidean. Now we require that Y be a compact topological manifold,

with or without boundary,

pathologies,

we also need conditions

are the topological a condition

equivalent

Q of a manifold

on the embedding

of those introduced

on a pair of manifolds

a submanifold

and that B* be a submanifold.

In order to rule out

of B’ in Y. These conditions

in [lo, Definition].

In that paper,

is defined which is called “neatly paired’. Recall that

M is called a neat submanifold

if aQ = Q n aM, and if

this intersection is transversal. (See [l 1, p. 301 for smooth manifolds. For PL manifolds, the same condition has been called 1ocalZyfEat in [20, p. 501. For either topological, PL or smooth manifolds, the same condition has been called clean, and stated in a slightly more general way which extends to manifolds with cubical comers in [17, pp. 12-131.) Note that the boundary

of a manifold

is never a neat submanifold.

So the class of manifold

pairs that we now define is broader than the class of pairs which consists of a manifold and a neat submanifold. It is equivalent in the topological setting to the smooth class used in [lo]. Definition

4.10. Let (M, Q) be a pair of spaces where A4 is a manifold

(with or without

boundary) and each component Qj of Q is a submanifold (with or without boundary). Suppose further that, for each j, either Qj c int M, or Qj c i3M, or Qj is a neat submanifold of M. Then (M, Q) is called a pair of manifolds where Q is neatly paired with M. If c E CIB,we write aB* for the component

of aB that contains

Lemma 4.11. Let f : (X, A) + (Y, B) where c E B. Suppose manifolds where B* is neatly paired with Y. (a) If p7’ (int B*) is connected,

then (pf, p;’ (B))

c. (Y, B’) is a pair of

is relatively

homogeneous

on

pT’(int B*). (b) If c E aB and pJ’(aB*)

is connected,

then

(& ,pj

’ (B)) is relatively homoge-

neous on pJ’(aB*). Proof. Since B* is neatly paired with Y and pf : pf --+ Y is a local homeomorphism, if follows that (Yf

,pj'(B*)) is a pair of manifolds, with or without boundary, where

p;’ (B’) is neatly paired with &. Let i, 2 E pJ’ (B*) be points that are either both in p-’ (int B’) of both in-p;’ of 6 and 6’ are in int(Yf)

(aB*). The definition of neatly paired implies that either both or both are in aYf. If 6,6’ E int(Yf),

we assume they are in a

neighborhood V in pf homeomorphic to R”. If they are in apf-,, then we suppose they are in a neighborhood V homeomorphic to

If V is homeomorphic to R”, the proof of Lemma 5.4 of [25] (see pp. 133-134) defines a continuous family of homeomorphisms 7t : pf + pf such that 70 is the identity and rl (i) = 6’. If i, 6’ E dpf, the functions ft : R” -+ R” defined on p. 133 of [25] take

RX Brown, H. Schimer / Topology and its Applications 92 (1999) 247-274

270

R;

to itself and therefore the definition

still gives us homeomorphisms

of ?f with the

required properties. Furthermore,

the points i,$

lie in the same neighborhood

As pJ’(B*)

is neatly paired with &,

neighborhood

with one of the three subsets Rk,

{cc= (21, . . .

)

z,)

E

l-2”:

21

=

of R” or R”,, where k is the dimension takes V i,$

V with R”

identifying

. . . =

Z,_&l

0,

=

of pj ’ (B’).

2,

2

B’)

Composing

the rt corresponding ‘1~~ : (?f, pJ’ (B*))

identity and ur (&) = &‘, so we have proved that onp;‘(intB*).

Ifp;‘(aB*)

is connected

find i)3 E ~7’ (CjB’) to demonstrate homogeneous

on p;l(aB*).

When c E B, the extension

+

o}

is connected

rt

and take any

B*) and choose points &= 61,

“‘, 6m =~onthearcsuchthat,foreachj=l,...,m-l,thepoints_~~and~~+l in some neighborhood Vj, homeomorphic to R” if pJ1 (int B’)

a family of homeomorphisms

this

Therefore the homeomorphism

B*). Connect 6 and 2 by an arc in p;‘(int

pJ’ (int B*) c auf.

or R”+ identifies

R$ or

n pJ’(B*) to itself. Now suppose that p;‘(int

E p;‘(int

V fl pT1 (B*) in pJ’(B*).

C

are or to RT if

int(Yf)

to these neighborhoods,

we define

(pf, pj’ (B*)) such that uo is the

(i;f,p;’

(B)) is relatively

homogeneous

and we choose b and &’in that set, then we can

in a similar manner that (pf, pj’ (B)) is relatively

0 of Theorem

1.2 to the relative

setting divides into two

cases, depending on whether or not B* has a boundary. We consider first the case in which the boundary of B* is nonempty. Recall that CIB* denotes the component of 3B that contains c. Theorem 4.12. Let f : (X, A) + (Y, B) be a map and choose c E B. Suppose (X, A*) is a pair of locally path-connected spaces, and (Y, B*) is a pair of manifolds where B* is neatly paired with Y. Assume further that B* is a manifold with boundary. Set Z = Za~e if c E aB and Z = 2~. otherwise.

If ~1(Y, c) = .F. Z, then

Nd(f; c) = 0 or Nd(f; c) = #(rl (Y, c)/F). Proof. Suppose connected

first that c E int B* and rrr(Y, c) = F . 1~~. Since S = int B* is Lemma 4.4 tell us that pfl (S) = pyl (int B*) is connected.

and 1 = IB’,

Therefore, since B’ is neatly paired with Y, Lemma 4.1 l(a) implies that (pf, pjl (B*)) is relatively homogeneous on pT’(int B*) and so in particular on p;‘(c). By Lemma 4.6, either all the root classes of f at c are relatively inessential or they are all relatively essential. Since B* has nonempty boundary, by [6, Theorem 7.41 we know that J: A* -+ B’ has no essential root classes, so N,l(f; c) = N+(f; c) which, by Lemma 4.6, can take only the values 0 and #(nr (Y, c)/F). The proof in the case c E aB* is the same, letting S = a B* in Lemma 4.4 and using case (b) of Lemma 4.11. •I

271

RF: Brown, H. Schimzer / Topology and ifs Applications 92 (1999) 247-274

The hypothesis be by-passed

of Theorem

rrr (Y, c) = 3.1~.

4.12 is satisfied when Y - B* can

because then ~1 (Y, c) = 1~~. For an example illustrating

which Y - B* cannot be by-passed,

Example 4.13. We retain X = Y = 5” x 5” and the embedding ii,(m)

Theorem 4.12 in

we may modify Example 4.9 as follows. i’ : 5” 4

Y such that

= (2m, m). However, now we let B be a closed regular neighborhood

of

i’(S’ )

and choose c E B. We use the same map f : X + Y, that is f(eiO, e’b) = (eki’, e’@) for some odd positive integer k, and we set f-’ (B) = A. Let i : B + Y be the inclusion, then i, : Z 4

Z $ Z such that &(m)

= (2m, m), as in Example

4.9. Therefore

the

computation in Example 4.9 proves in this case that rri (Y, c) = 3’zB* so the hypotheses of Theorem 4.12 are satisfied. Then either N,,r(f; c) = 0 or iV,,l(f; c) = #(rri (Y! c)/3) = k and since N(f; c) = k # 0, we conclude that N,,.l(f; c) = k. The statement of Theorem 4.12 includes the hypothesis rri (Y, c) = 3 . Z where the meaning of the symbol is Z = Zan* or Z = 1~. depending on the location of c. The following

example illustrates

this distinction.

Example

4.14. Let X’ be the one-point

union of an annulus

and a 2-disc and let A be

the boundary of X’, so it is the disjoint union of a circle and a figure-eight. Let X be constructed from X’ by removing two open discs from the interior of the annulus portion of X’. Let Y be the annulus

that we represent

as the set of points p in the plane with

1 < IpI < 4. Let B’ be the sub-annulus of points p such that 2 < IpI < 3, let D be an open disc in the interior of B’ and let B = B* = B’ - D, which is a submanifold of Y. We observe that 1,. = ~1 (Y, c), so the hypothesis rri (Y, c) = 3. Z is satisfied for any map f : (X, A) -+ (Y, B) and c E int B. Theorem 4.12 applies in this case and we always have N,l(f;c) = 0. Now take c E aD c aB, so i3B* = i3D, then Zas. is of rri (Y, c). Define f : (X, A) + (Y, B) on the punctured annulus portion of X to be a map of degree 2 onto B (the two discs deleted from X’ can be made the trivial subgroup to correspond boundary

to 0). Let the disc portion of X be mapped to B’ in such a way that its

is mapped homeomorphically

can be identified

onto the boundary

with 22, so the hypothesis

satisfied. It is not difficult to show that N,,l(f; #(V(Y,C)/3) case.

= 2, we see that the conclusion

of D. Note that 3.&n.

rrr (Y, c) = 3 . Z of Theorem c) # 0 and thus that N,,l(f; of Theorem

= 3

4.12 is not

c) = 1. Since

4.12 does not hold in this

We turn now to the case that the boundary of B* is empty. We shall consider the path-components of A3 of A* and the restrictions f, : A; -+ B* of f. For each pathcomponent A; such that N(fj ; c) # 0, we choose x3 E A: to be a root of fj at c and define IT3 to be the image of f,= : ~1 (AT. zj) + ~1 (B”, c). We denote the image of fx : ~1(X, xj) + TTI (Y, c) by 3j. We have observed that Lemma 4.4 implies that the condition ~1 (Y, c) = 3. Z is independent of the choice of the point x0 E f-‘(c) used to define 3. Thus the condition rri (Y, c) = 5 . Z B- will hold for some j if and only if it holds for all of them. We will write this condition as simply ~1 (Y; c) = 3.1~. _

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R.E Brown, H. Schirmer / Topology and its Applications 9.2 (1999) 247-274

Theorem 4.15. Let f : (X, A) -+ (Y, B) b e a map and choose c E B. Suppose (X, A*) is a pair of locally path-connected

spaces and (Y, B’)

B* is neatly paired with Y. Assume further ~1 (Y, c) = .F .I,.

where rf

, then

NT&f; c) = N(f; In particulal;

is a pair of manifolds

that B* is a manifold without boundary

if N(f;

c)

or

Nrel(f; c) = #(xl (Y, c)/F).

c) # 0, then N,,l(f;c)

= N(f;c).

Proof. We will again use the covering space pf : ?f + Y. By Lemma 4.4, the hypothesis 7ri (Y, c) = F. z p

and the connectedness

of B* imply that pyl (B*) is path-connected.

Therefore, since B* is neatly paired with Y, Lemma 4.1 l(a) implies that (?f, ~7’ (B)) is relatively

homogeneous

on all of p;’ (B’)

so, in particular,

it is relatively

homogeneous

on pyl (c) and we can apply Lemma 4.6 to conclude that either the root classes of f are all relatively inessential or they are all relatively essential. Since the space B’ is a manifold, Theorem 1.2 applied to a map f, : A5 + B* implies that either all the root classes of fj are essential

or all are inessential.

There are three cases to consider: (1) the root classes of f are all relatively inessential; (2) the root classes of f are all relatively essential and the root classes of fj are all inessential, for all j; (3) the root classes of f are all relatively essential for at least In case (I), clearly NTer(f; ? of f at c, with respect to of f, and so N,,l(f; c) =

essential

and the root classes of f, are all

one j. (2), the inverse f^-’ (c) of every factor c) = N(J; c). I n case h any lift f^ of f to Yf,must be a relatively essential root class N+(f; c) = #(xl (Y, c)/F). For the remaining case (3), we

have some j such that all the root classes of fT?are essential and rrt (Y, c) = $ . zB* . So if we can show that, for any root class R of f at c, there exists a root class ?? of f, at c contained

in R, then we can conclude

that in case (3) we have N+(f;

and therefore N,,I( f; c) = N(f; c). To obtain a suitable R, we will use as a tool the space p; :pj’(B*)

4

B* obtained

by restriction

c) = N+(f,

pJ’ (B*)

and the map

of pf : pf -+ Y. As ~1 (Y, c) = F.

it follows from Lemma 4.4 and [19, Lemma 2.1, p. 1501 that (pfl(B*),p’f) ing space of B’. Let i: B’ --f Y and l:pT’(B*)

+ pf be the inclusions.

base point for pf the point &J E pJ1 (c) which corresponds constant path at c. Then if follows from [ 19, Proposition 4 B* corresponds to the group space p; :pf’(B*)

f; c)

Zp,

is a coverWe select as

to the equivalence class of the 11.1, p. 771 that the covering

P;&Q($(B*),h)) = i,’ [p,?r(n,(Y;,h))] = i,‘(q). Now let pf; : SF, + B’ be the covering ~1 (B*,c). For kj : A; --+ X the inclusion,

space corresponding we have

to the subgroup Fj

of

R.R Brown, H. Schirmer / Topology and its Applications

We select as base point of gf,

92 (1999) 247-274

the point 4 which corresponds

to the equivalence

273

class

of the constant

path at c E b*. It follows from 119. Lemma 6.3, p. 1591 ^ that there ,. ^ -* - + pi’ (El*) such that hj(E,f) = & and p; 0 h, = pf;. exists a (unique) map h, : B Moreover, by [ 19, Lemma 6.; i0&:E;

~ --)

pf

p. 1601, (B^FJ, ij)

sends pi’(c)

1s ’ a covering space of pJ’ (B*). Hence

onto pi’(c).

Let R be a root class of f at c. Then R = f-’ where f^ is the lift off

and choose fJ : Aj ---) 6f iOPf3 = iop;oh3

(HIS)for some factor ?R E ~7’ (c),

with f^(zj) = &. We select any (‘j E pj’(c) 3

to be the lift of & : AT + B* with’fj(z,)

=pf

= CR,

= cj. As

o;oi,

and i o & (+) = 6, the map 3 o k,i is the unique lift of i o pf, : & maps the base points of Bj

with i:oitj(i”,)

and of Yf corresponding

3

+ Y to Yf which

to the constant path to each other.

Hence we can apply Lemma 4.1 to the map (X, A;) -+ (Y, B) defined by f, with the lift 1:o ij replacing 5. So we conclude 0 by R = fJ:‘(?;).

that ?? c R where z is the root class of fj given

Example 1.1, choosing c = 1, illustrates Theorem 4.15 with N,l(f; c) = N(J;c) = Idl. Example 2.3 does the same, when c E B, for the case of B disconnected. In Example 4.3, the set Y - B* = (S’ - (1)) x S’ cannot be by-passed. However, if we take a = 1 in that example then, as we observed following Lemma 4.4, for the Cartesian product space Y we have 7r’ (Y, c) = 3.1~. . Noting that N(f; c) = Jbl # 0, we conclude from Theorem 4.15 that N,l(f; c) = lb/. In our final example of the use of Theorem 4.15, the set Y - B* cannot be by-passed and we make use of the conclusion N,l(f; c) = #(7r’(Y. c)/F). Example 4.16. Let Y = S’ x S2 and B = S’ x { (1, 0, 0)}, and choose c E B. Let the embedding

i of B in Y be such that the image of i, : TT’(S’) 4 T’ (S’ x S2) can be

identified with 22. Since the image of i, is a proper subset of rr’ (Y), we conclude that Y - B* cannot be by-passed. Let g : S’ + S’ be a map of degree d, where d is odd. Let k:S’ x S’ + S2 be obtained by first mapping the figure-eight space that is the union of the generators of 7r’(S’ x 5”) to c, and then extending this map over the complement of the figure-eight space as a one-to-one map. So k is a map of degree 1. Now define f=gxI,_:X=S’x(S’xS’)+S’xS2=Y. Since we can identify

.F C TTI(Y, c) = Z with dZ, and d is relatively

prime to 2, we

see that the hypothesis rr’(Y, c) = .F. Z p is satisfied. We take A to be a single point of f-’ (B) so of course N(f; c) = 0. On the other hand, the map f : X 4 Y is of degree d # 0. so N(f; c) # 0 by Theorem 3.1 which implies that N,,l(f;c) is also nonzero, hence N,,l(f; c) = #(rr’(Y, c)/F) = jdj. Th’1s example satisfies the hypotheses of Theorem 3.3, so we conclude that N,,l(f; c) = IdI is sharp.

274

RX Brown, H. Schirmer / Topology and its Applications 92 (1999) 247-274

References [l] R. Brooks, Coincidences, roots and fixed points, Doctoral Dissertation, Univ. of California, Los Angeles (1967). [2] R. Brooks, Certain subgroups of the fundamental group and the number of roots of f(x) = a, Amer. J. Math. 95 (1973) 720-728. [3] R. Brooks, On the sharpness of the A2 and A, Nielsen numbers, J. Reine Angew. Math. 259 (1973) 101-108. [4] R. Brooks and C. Odenthal, Nielsen numbers for roots of maps of aspherical manifolds, Pacific J. Math. 170 (1995) 405-420. [5] R. Brown, A middle-distance look at root theory, Preprint. [6] R. Brown, B. Jiang and H. Schirmer, Roots of iterates of maps, Topology Appl. 66 (1995) 129-157. [7] R. Brown and H. Schirmer, Nielsen coincidence theory and coincidence-producing maps for manifolds with boundary, Topology Appl. 46 (1992) 65-79. [8] R. Brown and H. Schirmer, Correction to Nielsen coincidence theory and coincidenceproducing maps for manifolds with boundary, Topology Appl. 67 (1995) 233-234. [9] A. Dold, Lectures on Algebraic Topology, Grundlehren, Band 200, 2nd edn. (Springer, Berlin, 1980). [lo] R. Greene and H. Schirmer, Smooth realizations of relative Nielsen numbers, Topology Appl. 66 (1995) 93-100. [ 1 l] M. Hirsch, Differential Topology (Springer, New York, 1976). [12] H. Hopf, Zur Topologie der Abbildungen von Mannigfaltigkeiten II, Math. Ann. 102 (1930) 562-623. [ 131 C.G. Jang and S. Lee, A relative Nielsen number in coincidence theory, J. Korean Math. Sot. 32 (2) (1995) 171-181. [ 141 J. Jezierski, The Nielsen coincidence theory on topological manifolds, Fund. Math 143 (1993) 167-178. [15] J. Jezierski, A relative coincidence Nielsen number, Fund. Math. 149 (1996) 1-18. 1161 T.H. Kiang, The Theory of Fixed Point Classes (Springer-Verlag, Berlin/Science Press, Beijing, 1989). [ 171 R.C. Kirby and L.C. Siebenmann, Foundational Essays on Topological Manifolds, Smoothings and Triangulations, Annals of Mathematics Studies 88 (Princeton University Press, Princeton, NJ/University of Tokyo Press, Tokyo, 1977). [18] X. Lin, On the root classes of mapping, Acta Math. Sinica (N.S.) 2 (1986) 199-206. [19] W.S. Massey, Algebraic Topology: An Introduction (Harcourt, Brace and World, New York, 1967). [20] C. Rourke and B. Sanderson, Introduction to Piecewise-Linear Topology (Springer, Berlin, 1972). [21] H. Schirmer, A relative Nielsen number, Pacific J. Math. 122 (1986) 459473. [22] H. Schirmer, A survey of relative Nielsen fixed point theory, in: Proceedings of the Conference on Nielsen Theory and Dynamical Systems, Contemp. Math. 152 (1993) 291-309. [23] H. Schubert, Topology; English transl. by S. Moran (Allyn and Bacon, Boston, MA, 1968). [24] R. Skora, The degree of a map between surfaces, Math. Ann. 276 (1987) 415-423. [25] J.W. Vick, Homology Theory (Academic Press, New York, London, 1973). [26] K.-Y. Yang, A relative root Nielsen number, Comm. Korean Math. Sot. 11 (1) (1996) 245-252.